A uniform-density 6 kg disk of radius 0.3 m is mounted on a nearly frictionless axle. Initially it is not spinning. A string is wrapped tightly around the disk, and you pull on the string with a constant force of 25 N through a distance of 0.6 m. Now what is the angular speed?

A certain compound, A, reacts to form products according to the reaction $A \longrightarrow P$ The amount of A is measured as a function of time under a variety of different conditions and the tabulated results are shown here:

$$\begin{matrix} \text{Time (s)} & \ {25.0^{\circ} \mathrm{C} [A] (M)} & \ {35.0^{\circ} \mathrm{C} [A] (M)} & \ {45.0^{\circ} \mathrm{C} [A] (M)}\\ \text{0} & \text{1.000} & \text{1.000} & \text{1.000}\\ \text{10} & \text{0.779} & \text{0.662} & \text{0.561}\\ \text{20} & \text{0.591} & \text{0.461} & \text{0.312}\\ \text{30} & \text{0.453} & \text{0.306} & \text{0.177}\\ \text{40} & \text{0.338} & \text{0.208} & \text{0.100}\\ \text{50} & \text{0.259} & \text{0.136} & \text{0.057}\\ \text{60} & \text{0.200} & \text{0.093} & \text{0.032}\\ \end{matrix}$$

Have one group member make a graph of [A] vs. t, one group member make a graph of ln [A] vs. t, and one group member make a graph of 1/[A] vs. t using the data for 25 $^{\circ} \mathrm{C}$. Additional group members can make similar graphs for the other temperatures. What is the order of the reaction with respect to A? Explain your answer. a. Use the data to determine the rate constant at each temperature. b. What is the activation energy for this reaction? c. The same reaction is conducted in the presence of a catalyst, and the following data are obtained:

$$\begin{matrix} \text{Time (s)} & \ {25.0^{\circ} \mathrm{C} [A] (M)} & \ {35.0^{\circ} \mathrm{C} [A] (M)} & \ {45.0^{\circ} \mathrm{C} [A] (M)}\\ \text{0} & \text{1.000} & \text{1.000} & \text{1.000}\\ \text{0.1} & \text{0.724} & \text{0.668} & \text{0.598}\\ \text{0.2} & \text{0.511} & \text{0.433} & \text{0.341}\\ \text{0.3} & \text{0.375} & \text{0.291} & \text{0.202}\\ \text{0.4} & \text{0.275} & \text{0.190} & \text{0.119}\\ \text{0.5} & \text{0.198} & \text{0.122} & \text{0.071}\\ \text{0.6} & \text{0.141} & \text{0.080} & \text{0.043}\\ \end{matrix}$$

What effect does a catalyst have on the rate of the reaction? What is the activation energy for this reaction in the presence of the catalyst? How does it compare with the activation energy for the reaction when the catalyst isn't present?

Find the characteristic equation of the given symmetric matrix, and then by inspection determine the dimensions of the eigenspaces. (a)

$$\left[\begin{array}{ll}{2} & {6} \\ {6} & {18}\end{array}\right]$$

. (b)

$$\left[\begin{array}{rrr}{1} & {-2} & {2} \\ {-2} & {1} & {-2} \\ {2} & {-2} & {1}\end{array}\right]$$

. (c)

$$\left[\begin{array}{lll}{1} & {0} & {1} \\ {0} & {1} & {0} \\ {1} & {0} & {1}\end{array}\right]$$

. (d)

$$\left[\begin{array}{rrr}{2} & {2} & {2} \\ {2} & {2} & {-2} \\ {2} & {-2} & {2}\end{array}\right]$$

. (e)

$$\left[\begin{array}{llll}{0} & {3} & {0} & {0} \\ {3} & {0} & {0} & {0} \\ {0} & {0} & {0} & {3} \\ {0} & {0} & {3} & {0}\end{array}\right]$$

. (f)

$$\left[\begin{array}{rrrr}{0} & {0} & {2} & {-1} \\ {0} & {0} & {-1} & {2} \\ {2} & {-1} & {0} & {0} \\ {-1} & {2} & {0} & {0}\end{array}\right]$$

.